PHYSICAL REVIEW B 99, 144302 (2019)

Chaos-based Berry phase detector

Cheng-Zhen Wang,1 Chen-Di Han,1 Hong-Ya Xu,1 and Ying-Cheng Lai1,2,*

1School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA2Department of Physics, Arizona State University, Tempe, Arizona 85287, USA

(Received 12 December 2018; revised manuscript received 1 March 2019; published 10 April 2019)

The geometric or Berry phase, a characteristic of quasiparticles, is fundamental to the underlying quantummaterials. The discoveries of new materials at a rapid pace nowadays call for efficient detection of the Berryphase. Utilizing the α-T3 lattice as a paradigm, we find that, in the Dirac electron optics regime, the semiclassicaldecay of the quasiparticles from a chaotic cavity can be effectively exploited for detecting the Berry phase. Inparticular, we demonstrate a one-to-one correspondence between the exponential decay rate and the geometricphase for the entire family of α-T3 materials. This chaos-based detection scheme represents an experimentallyfeasible way to assess the Berry phase and to distinguish the quasiparticles.

DOI: 10.1103/PhysRevB.99.144302

I. INTRODUCTION

The geometric phase, commonly referred to as thePancharatnam-Berry phase or simply the Berry phase, is afundamental characteristic of the quasiparticles of the under-lying quantum material. When a system is subject to a cyclicadiabatic process, after the cycle is completed, the quantumstate returns to its initial state except for a phase difference—the Berry phase [1–3]. In general, the exact value of the Berryphase depends on the nature of the quasiparticles and hencethe underlying material. For example, the Berry phases inmonolayer graphene [4,5] and graphite bilayers [6] are ±π

and 2π , respectively. In α-T3 lattices, for different values ofα, the Berry phases associated with the quasiparticles aredistinct [7].

Advances in physics, chemistry, materials science, andengineering have led to the discoveries of new materials at anextremely rapid pace, e.g., the various two-dimensional Diracmaterials [8–10]. These materials host a variety of quasiparti-cles with distinct physical characteristics, including the Berryphase. The ability to detect the Berry phase for a new ma-terial would generate insights into its physical properties forpotential applications. Conventionally, this can be done usingthe principle of Aharonov-Bohm interference. For example,an atomic interferometer was realized in an optical latticeto directly measure the Berry flux in momentum space [11].Graphene resonators subject to an external magnetic fieldcan be used to detect the Berry phase [12,13]. Specifically,for a circular graphene p-n junction resonator, as a resultof the emergence of the π Berry phase of the quasiparticles(Dirac fermions) when the strength of the magnetic field hasreached a small critical value, a sudden and large increasein the energy associated with the angular momentum statescan be detected. In photonic crystals, a method was proposedto detect the pseudospin-1/2 Berry phase associated with theDirac spectrum [14]. In such a system, the geometric Berry

*ying-cheng.lai@asu.edu

phase acquired upon rotation of the pseudospin is typicallyobscured by a large and unspecified dynamical phase. It wasdemonstrated [14] that the analogy between a photonic crystaland graphene can be exploited to eliminate the dynamicalphase, where a minimum in the transmission arises as a directconsequence of the Berry phase shift of π acquired by acomplete rotation of the pseudospin about a perpendicularaxis.

In this paper, we report a striking phenomenon in two-dimensional Dirac materials, which leads to the principle ofchaos-based detection of the Berry phase. To be concrete, weconsider the entire α-T3 material family. An α-T3 material canbe synthesized by altering the honeycomb lattice of grapheneto include an additional atom at the center of each hexagon,which, for α = 1, leads to a T3 or a dice lattice that hostspseudospin-1 quasiparticles with a conical intersection oftriple degeneracy in the underlying energy band [15–41].An α-T3 lattice is essentially an interpolation between thehoneycomb lattice of graphene and a dice lattice, where thenormalized coupling strength α between the hexagon andthe central site varies between 0 and 1 [7,42–46], as shownin Fig. 1(a). Theoretically, pseudospin-1 quasiparticles aredescribed by the Dirac-Weyl equation [16,17,34]. Supposewe apply an appropriate gate voltage to generate an externalelectrostatic potential confinement or cavity of an α-T3 lat-tice. The mechanism for Berry phase detection arises in theshort wavelength or semiclassical regime, where the classicaldynamics are relevant and can be treated according to rayoptics with reflection and transmission laws determined byKlein tunneling, the theme of the emergent field of Diracelectron optics (DEO) [13,47–75]. If the shape of the cavity ishighly symmetric, e.g., a circle, the classical dynamics of thequasiparticles are integrable. However, if the cavity bound-aries are deformed from the integrable shape, chaos can arise.We focus on the energy regime V0/2 < E < V0 in which Kleintunneling is enabled, where V0 is the height of the potential[Fig. 1(b)], so that the relative effective refractive index ninside the cavity falls in the range [−∞,−1]. As a result, thereexists a critical angle for total internal reflections. For different

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WANG, HAN, XU, AND LAI PHYSICAL REVIEW B 99, 144302 (2019)

FIG. 1. Schematic illustration of an α-T3 cavity and the energydispersion relation. (a) α-T3 lattice structure. (b) The electron andhole energy dispersion relations in different spatial regions. (c) Apossible scheme of experimental realization of the cavity throughan applied gate voltage. The amount of the voltage is such that thequasiparticles are in the Klein tunneling regime.

values of the material parameter α, the physical characteristicsof the quasiparticles, in particular the values of the Berryphase, are different. Our central idea is then that, for a fixedcavity shape, the semiclassical decay laws for quasiparticlescorresponding to different values of α would be distinct. Ifthe classical cavity dynamics contain a regular component,the decay laws will be algebraic [76–80], but we find thatthe differences among them will not be statistically significantenough to allow lattices of different values of α to be distin-guished. However, when the cavity is deformed so that theclassical dynamics are fully chaotic, the decay law becomesexponential [81]. The striking phenomenon is that the expo-nential decay rate for different values of α can be statisticallydistinguished to allow the Berry phase of the quasiparticles tobe unequivocally detected, leading to the birth of chaos-basedBerry phase detectors. We note that in microcavity optics,classical chaos can be exploited to generate lasing with a highquality factor and good emission directionality at the sametime [82–92].

II. HAMILTONIAN AND DIRAC ELECTRON OPTICS

The α-T3 lattice system has the advantage of generatinga continuous spectrum of quasiparticles with systematicallyvarying Berry phase through the tuning of the value of theparameter α in the unit interval. At the two opposite ends ofthe spectrum, i.e., α = 0, 1, the quasiparticles are pseudospin-1/2 Dirac fermions and pseudospin-1 Dirac-Weyl particles,respectively. As illustrated in Fig. 1, the lattice has threenonequivalent atoms in one unit cell, and the interactionstrength is t between A and B atoms and αt between B and Catoms, where t is the nearest-neighbor hopping energy of thegraphene lattice. A cavity of arbitrary shape can be realizedby applying an appropriate gate voltage through the STMtechnique [13,60,93], as shown in Fig. 1(c). We consider cir-cular and stadium-shaped cavities that exhibit integrable andchaotic dynamics, respectively, in the classical limit [94]. Thelow-energy Hamiltonian for the α-T3 system about a K pointin the hexagonal Brillouin zone is [42,45] H = Hkin + V (x)I ,where Hkin is the kinetic energy, V (x) is the applied potential

that forms the cavity, and I is the 3 × 3 identity matrix. Thecoupling strength α can be conveniently parameterized asα = tan ψ . The kinetic part of the rescaled Hamiltonian (bycos ψ) is

Hkin =⎡⎣ 0 fk cos ψ 0

f ∗k cos ψ 0 fk sin ψ

0 f ∗k sin ψ 0

⎤⎦, (1)

where fk = vF (ξkx − iky), vF is the Fermi velocity, k =(kx, ky) is the wave vector, and ξ = ± are the valley quan-tum numbers associated with K and K ′, respectively. In thesemiclassical regime where the particle wavelength is muchsmaller than the size of the cavity so that the classical dynam-ics are directly relevant, the DEO paradigm can be instatedto treat the particle escape problem, which is analogous tothe decay of light rays from a dielectric cavity. In DEO, theessential quantity is the transmission coefficient of a particlethrough a potential step, which can be obtained by wavefunction matching as [45]

T = 4ss′ cos θ cos φ

2 + 2ss′ cos (θ + φ) − sin2 2ψ (s sin θ − s′ sin φ)2,

(2)

where s = ± and s′ = ±, with the plus and minus signsdenoting the conduction and valence bands, respectively, andincident and transmitted angles are φ and θ , respectively.Imposing conservation of the component of the momentumtangent to the interface, we get

sin θ = (E/|E − V0|) sin φ.

(More details about electron transmission through a potentialstep can be found in Appendix A.) Our focus is on the survivalprobability of the quasiparticles from an α-T3 cavity for theentire material spectrum: 0 � α � 1.

We set the amount of the applied voltage such that the en-ergy range of the quasiparticles is V0/2 < E < V0 (the Kleintunneling regime). In the optical analog, the correspondingrelative effective refractive index inside the cavity is n =E/(E − V0), and that outside of the cavity is n = 1. Due toKlein tunneling, the range of relative refractive index in thecavity is negative: −∞ < n < −1. As a result, a critical angleexists for the tunneling of electrons through a simple staticelectrical potential step, which is sin φc = (V0 − E )/E and isindependent of the α value [45]. This behavior is exemplifiedin the polar representation of the transmission in Fig. 2(a),which shows that the value of the transmission increases withα. As the value of α is varied in the unit interval, the criticalangle remains unchanged.

III. RESULTS

A. Algebraic decay of α-T3 quasiparticles from a circular(integrable) cavity

The classical phase space contains Kolmogorov-Arnold-Moser tori and an open area through which particles (rays)

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FIG. 2. Semiclassical decay of quasiparticles from a cavity inan α-T3 lattice. For particle energy E = 0.53V0 (within the Kleintunneling regime) and relative refractive index n = −1.1277 insideof the cavity, (a) transmission T across a potential step as a functionof incident angle φ for a number of equally spaced α values.(b) SPTD for the circular (integrable) cavity on a double-logarithmicplot, where the blue circles, red squares, orange diamonds, purpleupward triangles, and green downward triangles are numerical resultsfor the five α values in (a) and the solid lines are the theoreticalpredictions. The decay is algebraic, but the decay exponent is aconstant independent of the value of α. (c) SPTD for a stadium-shaped (chaotic) cavity of semicircle radius 1 and straight edge oflength 2 on a semilogarithmic plot. The color legends are the sameas in (b). In this case, the decay is exponential, and its rate dependson the value of α. Measuring the exponential decay rate then givesthe value of α and the corresponding Berry phase of the underlyingmaterial lattice system.

escape. Initializing an ensemble of particles (e.g., 107) in theopen area, the survival probability time distribution (SPTD) is

given by

Psv (t ) =∫ L

0ds

∫ pc

−pc

d pI (s, p)R(p)N (t ), (3)

where L is the boundary length, pc = sin φc = 1/|n|, with φc

being the critical angle for total internal reflection, R(p) =1 − T is the reflection coefficient for the α-T3 quasiparticleswith transmission T defined in Eq. (2), N (t ) = t/(2 cos φ) isthe number of bounces off the boundary, and I (s, p) = |n|/2Lis the uniform initial distribution.

Consider a circle of unit radius. Using the length of the raytrajectory as the timescale, we can rewrite Eq. (3) as

Psv (t ) = |n|∫ φc

0dφ cos φ exp

[− t

2 cos φln

(1

R

)], (4)

with

R−1 = 1 + −4 cos θ cos φ

2 + 2 cos (θ − φ) − sin2 2ψ (sin θ + sin φ)2.

(5)

The behavior of the particle transmission coefficient shownin Fig. 2(a) indicates that particles near the critical angle φc

can survive for a longer period of time in the cavity. We canthen expand the ln( 1

R ) term about the critical angle φc bydefining a new variable χ with φ = φc − χ and exploiting theapproximation χ → 0. We have

ln

(1

R

)≈ 4

√2|n| cos φc cos φc

2 + [2|n| − sin2 2ψ (|n| + 1)2 sin2 φc]χ1/2. (6)

Substituting Eq. (6) into Eq. (4), we obtain the SPTD as

Psv = 1

4t−2

{2 + [2|n| − sin2 2ψ (|n| + 1)2]

1

|n|2}2

= C(n, ψ )t−2. (7)

This indicates that the quasiparticles decay algebraically fromthe cavity and the value of the decay exponent is 2, regardlessof the value of α. For certain values of |n|, as the value of α

changes from 0 to 1, the decay coefficient C(n, ψ ) decreases,as shown in Fig. 2(b). Here, SPTD for the circular cavity iscalculated with 107 random initial points in the open region ofthe phase space. The trajectory from each point is traced withthe reflection coefficient R(p) at the boundary. The survivalprobability between t and t + with = 1 is calculated withan initial probability of 1 at t = 0. From Fig. 2(b), we seethat both theoretical and numerical results show an algebraicbehavior in the long-time regime with an exponent of 2.

Experimentally, to distinguish the nature of the quasipar-ticles and to detect the Berry phase, the decay coefficient isnot a desired quantity to measure as it reflects the short-timebehavior of the decay process. In fact, it depends not only onthe nature of the material (as determined by the value of α) butalso on the detailed system design. The long-time behavior ofthe decay is characterized by the algebraic decay exponent,which does not depend on the details of the experimentaldesign, and hence, it can possibly be exploited for Berry phasedetection. However, for an integrable cavity, the algebraicdecay exponent remains constant as the value of α is changed,as shown in Fig. 2(b). It is thus not feasible to distinguish

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the quasiparticles by their long-time behavior, ruling outintegrable cavities as a potential candidate for detecting theBerry phase.

B. Exponential decay of α-T3 quasiparticlesfrom a chaotic cavity

For the stadium cavity, the classical dynamics are chaotic,leading to random changes in the direction of the propagatingray. In this case, the survival probability of the quasiparticlesin the cavity decays exponentially with time, as shown inFig. 2(c), where the long-time behavior is determined bythe exponential decay rate. The striking phenomenon is thatthe decay rate increases monotonically as the value of thematerial parameter α is increased from 0 to 1, suggesting thepossibility of using the exponential decay rate to distinguishthe α-T3 materials and to detect the intrinsic Berry phase.The difference in the decay rate can be further demonstratedby calculating its dependence on the absolute value |n| fordifferent values of α, as shown in Fig. 3(a). For small valuesof |n|, the difference in the decay rate is relatively large,indicating a stronger ability to discern the α-T3 quasiparticles.For large values of |n|, the difference in the decay rate issomewhat reduced. This is expected because, as the value of|n| is increased from 1, the transmission for the materials atthe two ends of the α-T3 spectrum, namely, graphene and apseudospin-1 lattice, decreases continuously. For |n| → ∞,the transmission tends to zero. This result indicates that theoptimal regime to discern the quasiparticles for α-T3 occursfor |n| above 1 but not much larger, corresponding to theregime where the particle energy is slightly above half of thepotential height.

In general, for a given value of α, the exponential decayrate is inversely proportional to n, which can be argued as fol-lows [95,96]. For Psv (t ) ∼ exp (−γ t ), we have dPsv (t )/dt ∼−γ Psv (t ) ∼ −[〈T (p)〉/〈d〉]Psv (t ), where 〈T (p)〉 and 〈d〉 arethe average transmission and the distance between two con-secutive collisions in the chaotic cavity, respectively. The de-cay rate can then be obtained in terms of the steady probabilitydistribution Ps(s, p) as

γ = 〈T (p)〉/〈d〉 = 〈d〉−1∫ L

0ds

∫ 1

−1d pPs(s, p)T (p). (8)

In the Klein tunneling regime V0/2 < E < V0 (−∞ < n <

−1), we can derive an analytical expression for the expo-nential decay rate based on a simple model of the steadyprobability distribution (SPD) for the stadium-shaped cavitythat generates fully developed chaos in the classical limit [95].Specifically, we assume that the SPD is a uniform distributionover the whole phase space except the open regions relatedto the linear segments of the stadium boundary. The decayrate can then be expressed in terms of the steady probabilitydistribution:

γ = 2πR

2(πA/L)(L − 2l/|n|)∫ 1/|n|

−1/|n|d pT (p), (9)

where T (p) is the transmission coefficient defined inEq. (2) and the average path length of ray trajectory seg-ments between two successive bounces is 〈d〉 = πA/L,with A = πR2 + 2Rl and L = 2πR + 2l being the area

FIG. 3. Dependence of the semiclassical exponential decay ratefrom a chaotic cavity on the effective refractive index and thedetection of the Berry phase. (a) For α = 0, 0.25, 0.5, 0.75, 1, thedecay rate versus the refractive index, where the blue circles, redsquares, orange diamonds, purple upward triangles, and green down-ward triangles are the respective numerical results and the dashedcurves are theoretical predictions. (b) For E/V0 = 0.53, detection ofBerry phase (red squares) based on the decay rate (blue circles).As the value of α is changed from 0 to 1, there is a one-to-onecorrespondence between the exponential decay rate and the Berryphase.

and boundary length of the stadium, respectively. Substi-tuting the expressions sin θ = |n|p, cos θ = −

√1 − sin2 θ =

−√

1 − n2 p2, sin φ = p, and cos φ =√

1 − p2 into the ex-pression of T (p), we get

T = 4√

1 − p2√

1 − n2 p2/[2 + 2√

1 − p2√

1 − n2 p2

+ 2|n|p2 − sin2 2ψ (n2 p2 + p2 + 2|n|p2)]. (10)

In the limit |n| ≈ 1, imposing a change of variable x = n2 p2

to get d p = dx/(2|n|√x), we can write the decay rate in termsof variable x as

γ = 2πR

2(πA/L)(L − 2l/|n|)∫ 1

0

dx√x

T (x)

= 2πR

2(πA/L)(L − 2l/|n|)∫ 1

0

dx√x

(1−x)[1−(sin2 2ψ )x]−1

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= 2πR

2(πA/L)(L − 2l/|n|)B(1/2, 2)F (1, 1/2; 5/2; sin2 2ψ )

≈ 2πR

2(πA/L)(L − 2l/|n|)4

3

(1 + 1

5sin2 2ψ + · · ·

), (11)

where B(x, y) = �(x)�(y)/�(x + y) is the beta function andF (α, β; γ ; z) is the Gauss hypergeometric function.

In the |n| � 1 regime, we use the change in variable x =np to simplify the decay rate integral. The decay rate becomes

γ = 4πR

2(πA/L)L|n|∫ 1

0

4√

1 − x2

2 + 2√

1 − x2 − sin2(2ψ )x2, (12)

which is inversely proportional to the absolute value of therefractive index |n|. More importantly, the decay rate dependson the material parameter α monotonically (α = tan ψ , withα increasing from 0 to 1). We note that the theoretical resultsin Fig. 2(c) are obtained by using the integration formula (9)directly. The approximation used to derive Eqs. (11) and (12)is to facilitate an analytic demonstration of the scaling of thedecay rate with n. The formulas also reveal that the decay rateincreases monotonically with α.

Numerically, we choose the stadium shape with the semi-circle radius to be 1 and the length of the straight long edgeto be 2. In the calculation, we use a random ensemble of 107

initial points spread over the whole phase space and trace thesurvival probability with time, which is scaled by the lengthof trajectory as in the case of a circular cavity. The numericalresults are consistent with the theoretical cases based on SPDapproximation, as shown in Fig. 2(c).

C. Detection of Berry phase

The Berry phase associated with an orbit in the conicalbands is given by [7]

φBξ = πξ cos (2ψ ) = πξ

(1 − α2

1 + α2

). (13)

For the flat band, the Berry phase is

φB0,ξ = −2πξ cos (2ψ ) = −2πξ

(1 − α2

1 + α2

). (14)

We take ξ = ±1 for the K and K ′ valleys, respectively. Forξ = 1, the dependence of the Berry phase on α is shown inFig. 3(b). As the value of α is increased from 0 to 1, the Berryphase decreases monotonically from π to 0. At the same time,the exponential decay rate increases monotonically. There isthen a one-to-one correspondence between the decay rate andthe Berry phase for the entire spectrum of α-T3 materials,justifying a semiclassical chaotic cavity as an effective Berryphase detector.

IV. DISCUSSION

To summarize, we uncovered a phenomenon in relativisticquantum chaos that can be exploited to detect the Berryphase of two-dimensional Dirac materials. In particular, forthe spectrum of α-T3 materials, in the semiclassical regime,the decay of the quasiparticles from a chaotic cavity dependson the intrinsic material parameter. Experimentally, the cavitycan be realized through a gate voltage, where, locally, the

boundary of the cavity is effectively a potential step. When theFermi energy of the quasiparticles is above half but below thepotential height, the system is in the Klein tunneling regime,rendering applicable Dirac electron optics. In this case, therelative effective refractive index inside the cavity is between−∞ and −1, so a critical angle exists for the semiclassicalray dynamics. Because of the close interplay between Kleintunneling and the value of the Berry phase, measuring thequasiparticle escape rate leads to direct information about theBerry phase and for differentiating the α-T3 materials. Ouranalysis and calculation have validated this idea: we haveindeed found a one-to-one correspondence between the expo-nential decay rate and the value of the Berry phase. In termsof basic physics, our finding builds up a connection betweenclassical chaos and Berry phase. From an applied standpoint,because of the fundamental importance of the Berry phase indetermining the quantum behaviors and properties of mate-rials, our work, relative simplicity notwithstanding, providesan effective and experimentally feasible way to assess theBerry phase for accurate characterization of the underlyingmaterial. This may find broad applications in materials sci-ence and engineering, where new nanomaterials are beingdiscovered at a rapid pace, demanding effective techniques ofcharacterization.

A possible experimental scheme to detect the Berry phasefor the family of α-T3 materials is as follows. For each type ofmaterial, one first makes a chaotic cavity (e.g., a stadium- ora heart-shaped domain). One then measures the quasiparticledecay rate for the graphene cavity (corresponding to α = 0).Since the Berry phase of graphene is known, one can use themeasurement as a baseline for calibrating the results fromother materials in the family. Finally, making use of theone-to-one correspondence between the curves of the decayrate and the Berry phase versus the material parameter α astheorized in this paper, one can detect the actual Berry phasefor the material with any value of α for 0 < α � 1.

ACKNOWLEDGMENT

We would like to acknowledge support from the VannevarBush Faculty Fellowship program sponsored by the BasicResearch Office of the Assistant Secretary of Defense forResearch and Engineering and funded by the Office of NavalResearch through Grant No. N00014-16-1-2828.

APPENDIX A: BAND STRUCTURE AND WAVE VECTORSACROSS A POTENTIAL STEP

To better understand the optical-like decay behavior ofquasiparticles from a cavity formed by an electrostatic gatepotential, we illustrate the electron band structure and thewave vectors across a potential step associated with a trans-mission process, as shown in Fig. 4. We also indicate aclassification scheme of the regimes with different valuesof the refractive index, which are determined by differentvalues of the applied potential relative to the Fermi energy. Inparticular, there are regimes of positive and negative values ofthe refractive index with respect to cases where a critical angleexists or is absent. For convenience, the incident electron is

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FIG. 4. Schematic illustration of the band structures and wave vectors across a potential step with different values of the refractive index.(a)–(d) Band structures across the potential step with different values of the gate potential (corresponding to different values of the refractiveindex) at fixed Fermi energy. The black arrows denote the wave vector directions (only the cases with the wave vector in the x direction areshown). In the regime of negative refractive index, the wave vector directions are reversed. (e)–(h) Electron wave vectors with the incident andtransmitted angles φ and θ , respectively.

assumed to be in the conduction band, i.e., with a positiveFermi energy, and we vary the potential height V0. WhenV0 is larger than the Fermi energy, the transmitted electronis in the valence band. In this case, the wave vector has anegative x and a positive y component, but the direction ofthe velocity remains unchanged, leading to a negative valueof the refractive index.

More specifically, for gate potential height in the rangeV0/2 < E < V0, the value of the refractive index n = E/(E −V0) falls in the range −∞ < n < −1. There is a critical anglein this case, which is determined by sin θ = 1 = (E/|E −V0|) sin φc. The transmission angle can be obtained in termsof incident angle φ as

θ = π − tan−1 (sin φ)(E/V0)√(1 − E/V0)2 − [(sin φ)(E/V0)]2

= π + tan−1 n sin φ√1 − (n sin φ)2

, (A1)

where the relations sin θ = (E/|E − V0|) sin φ and cos θ =−

√1 − sin2 θ have been used. The band structure and angles

corresponding to the wave vectors are shown in Figs. 4(a)and 4(e), respectively.

In the regime where the potential height satisfies 0 < E <

V0/2, the value of the refractive index is in the range −1 <

n < 0. As a result, there is no critical angle. The transmissionangle can be obtained in the same form as Eq. (A1). Aschematic illustration of the band structure and the wavevector angles for this case are shown in Figs. 4(b) and 4(f),respectively.

For V0 < 0 < E , the value of the refractive index is inthe positive range, 0 < n < 1, because both the incident andtransmitted electrons are in the conduction band. There isno critical angle in this case. The transmission angle can be

obtained as

θ = tan−1 (sin φ)(E/V0)√(1 − E/V0)2 − [(sin φ)(E/V0)]2

= tan−1 n sin φ

1 − (n sin φ)2. (A2)

where the relations sin θ = [E/(E − V0)] sin φ and cos θ =√1 − sin2 θ are used. The band structure and wave vectors’

related angles are depicted in Figs. 4(c) and 4(g), respectively.In the regime 0 < V0 < E , the refractive index is in the

range 1 < n < ∞ with both the incident and transmittedelectrons in the conduction band. There is a critical angle inthis case determined by sin θ = 1 = [E/(E − V0)] sin φc. Thetransmission angle can be obtained in the same form as inEq. (A2). The band structures and wave vectors are illustratedin Figs. 4(d) and 4(h), respectively.

APPENDIX B: SURVIVAL PROBABILITY DISTRIBUTIONOF α-T3 QUASIPARTICLES IN DIFFERENT

ENERGY REGIMES

For completeness, we derive the decay law of the survivalprobability of α-T3 quasiparticles and obtain the decay ratein other energy regimes. We argue that the decay law inthese regimes is practically infeasible for detecting the Berryphase. For example, in the regimes where there is no criticalangle, the decay can be too fast for it to be useful. In theregimes where there is a critical angle, the decay for distinctquasiparticles from the material family follows a similar law,making it difficult to distinguish the different quasiparticles.

1. The 0 < E < V0/2 regime

In this energy regime, the refractive index n = E/(E − V0)of the cavity is in the range −1 < n < 0. In this regime, thereexists no critical angle for rays inside the cavity. Figure 5(a)

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FIG. 5. Survival probabilities from integrable and chaotic cavities for 0 < E < V0/2. For E/V0 = 1/3 and n = −0.5, (a) transmissionversus the incident angle on a polar plot, (b) decay of the survival probability from a circular (integrable) cavity with time, and (c) decay of thesurvival probability from a stadium-shaped (chaotic) cavity.

shows that the transmission is nonzero for all angles and itincreases with decreasing α values. In this case, the decay ofquasiparticles is exponential, and it does not depend on thenature of the classical dynamics, i.e., integrable or chaotic, asshown in Fig. 5.

A theoretical explanation of the features in Fig. 5 is asfollows. Due to the absence of a critical angle for Diracelectron optical rays in the energy range 0 < E < V0/2, thesurvival probability from a circular (integrable) cavity ismainly determined by the ray behavior about φ = π/2. Let-ting φ = π/2 − x, where x is a small angle deviation fromπ/2, and using the approximations

sin φ ≈ sin φc − (cos φc)x,

cos φ ≈ cos φc + (sin φc)x,

sin θ ≈ |n|[sin φc − (cos φc)x],

cos θ ≈ −√

1 − n2[sin φc − (cos φc)x]2,

we get

ln R−1 = 4x√

1 − n2

2 + 2|n| − sin2 (2ψ )(1 + |n|)2, (B1)

where R = 1 − T , with T being the transmission coefficientdefined in Eq. (2) in the main text. The survival probabilitycan be expressed as

Psv = exp

{− 2

√1 − n2

2 + 2|n| − sin2 (2ψ )(1 + |n|)2t

}(B2)

For a chaotic cavity, the angle distribution is random,leading to an exponential behavior of the survival probability.We can obtain the expression for the decay rate γ by approxi-mating Psv as

Psv (t ) ≈ 〈1 − T (p)〉t/〈d〉 = exp {ln [1 − 〈T (p)〉](t/〈d〉)}.(B3)

The decay rate can be expressed as

γ = − 1

〈d〉 ln [1 − 〈T (p)〉]. (B4)

For either the integrable or the chaotic cavity, the exponentialdecay rate depends on the material parameter α, which, inprinciple, can be used to detect the Berry phase. However, due

to the lack of a critical angle in this energy range, experimen-tally, it would be difficult to confine the quasiparticles. Indeed,compared with the exponential decay from a chaotic cavity inthe Klein tunneling regime (V0/2 < E < V0) as treated in themain text, here, the decay is much faster.

2. The 0 < V0 < E regime

For the energy range 0 < V0 < E with the refractive indexn = E/(E − V0) of the cavity in the range 1 < n < ∞, thesurvival probability with time exhibits an algebraic decayfrom an integrable cavity and an exponential decay from achaotic cavity, which is characteristically similar to the decaybehaviors in the Klein tunneling regime (V0/2 < E < V0)treated in the main text. A difference is that, for 0 < V0 < E ,the dependence of the transmission on the material parameterα is much weaker in the sense that, as the value of α isincreased from 0 to 1, the transmission barely changes. Itis thus practically difficult to distinguish the quasiparticlesfor different materials. These behaviors are shown in Fig. 6,where the analytical fitting is calculated in the same way as inthe main text.

3. The V0 < 0 < E regime

In the energy regime V0 < 0 < E with the refractive indexn = E/(E − V0) of the cavity in the range 0 < n < 1, thedecay of the survival probability is similar to that in the0 < E < V0/2 regime. In particular, regardless of the natureof the classical dynamics (integrable or chaotic), the survivalprobability exhibits an exponential decay with time, as shownin Fig. 7. Again, compared with the energy regime of Kleintunneling, the decay is much faster here, making experimentaldetection of the Berry phase difficult.

APPENDIX C: COMPARISON BETWEEN THE DECAYBEHAVIORS FOR PSEUDOSPIN-1/2 and PSEUDOSPIN-1

QUASIPARTICLES

The best-studied material in the α-T3 family is graphene,corresponding to α = 0 [97]. There is also growing interestin the material at the other end of the spectrum: α = 1, forwhich the quasiparticles are of the pseudospin-1 nature. We

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FIG. 6. Survival probabilities from integrable and chaotic cavities for 0 < V0 < E . For E/V0 = 8.8309 (n = 1.1277), (a) polar representa-tion of the transmission with respect to the incident angle, (b) decay with time of the survival probability from a circular (integrable) cavity,and (c) decay of survival probability from a stadium-shaped (chaotic) cavity.

offer a comparison of the decay behaviors of the quasiparticlesat these two extreme cases.

In the energy range 0 < E < V0/2 with Klein tunneling[corresponding to the negative refractive index: −1 < n =E/(E − V0) < 0], there is no critical angle for total inter-nal reflection. For both integrable and chaotic cavities, thesurvival probability decays exponentially with time, with noqualitative difference. As the absolute value of the refractiveindex is increased, the range of angle for transmission islarge for pseudospin-1 quasiparticles, but the range is smallerfor pseudospin-1/2 quasiparticles. For integrable cavities, thedifference is somewhat larger.

In the energy range for Klein tunneling: V0/2 < E < V0

(−∞ < n < −1), a critical angle arises, above which thereare total internal reflections. For an integrable cavity, the sur-vival probability decays algebraically with time, but the decayis exponential for a chaotic cavity. In the integrable case,the algebraic decay exponents have approximately identicalvalues for the pseudospin-1 and pseudospin-1/2 particles.However, for a chaotic cavity, the decay of pseudospin-1quasiparticles is much faster than that of pseudospin-1/2quasiparticles. Chaos can thus be effective in detecting theBerry phase to distinguish the two types of quasiparticles.In fact, as demonstrated in the main text, chaos in the

Klein tunneling regime can be effective for detecting theBerry phase across the entire material spectrum of the α-T3

family.In the energy range of V0 < E (1 < n < ∞), a critical

angle exists. The decay behavior of the survival probability isalgebraic for an integral cavity and exponential for a chaoticcavity. The difference in the transmission versus the incidentangle is small for pseudospin-1 and pseudospin-1/2 quasi-particles, leading to a similar value of the algebraic decaycoefficient in the integrable case and a similar exponentialdecay law in the chaotic case. In this energy range, usingthe decay behavior to discern the quasiparticles would bepractically difficult.

In the energy range V0 < 0 < E (0 < n < 1), there is nocritical angle, and the decay behavior is exponential for bothintegrable and chaotic cavities. As the energy is increased,the difference in the decay behaviors of pseudospin-1 andpseudospin-1/2 quasiparticles diminishes, ruling out the pos-sibility of exploiting the decay for detection of Berry phase.

Finally, we note a symmetry-related phenomenon: for spin-1 quasiparticles the behaviors of the survival probability areidentical for positive and negative refractive index regimesas a result of symmetry in the expression of the transmissioncoefficient.

FIG. 7. Survival probability from integrable and chaotic cavities in the V0 < 0 < E energy regime. For E/V0 = −1 (n = 0.5), (a) polarrepresentation of the transmission versus the incident angle, (b) decay of survival probability from a circular (integrable) cavity, and (c) decayof survival probability from a stadium-shaped (chaotic) cavity.

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